Daytime temperatures sit around 28°C only really dropping to 15°C in the evenings, so this is still relatively agreeable and you will only need fairly lightweight layers to keep you warm during morning and evening game drives.
If you want to combine your safari holiday with some trekking such as Mount Kilimanjaro or Mount Meru, June to October is a good time to go, but remember that the mountain tops remain snow-capped year-round and temperatures will be low.
The east coast hosts the Bagamoyo Arts Festival in September celebrating with traditional dance, music and performances as well as art exhibitions and workshops.